Euclid's lemma

In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: If a prime divides the product of two numbers, it must divide at least one of those numbers. It is also called Euclid's first theorem[1][2] although that name more properly belongs to the side-angle-side condition for showing that triangles are congruent.[3] For example, 133 × 143 = 19019, and since 19019 is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In fact, 133 ÷ 19 = 7.

The usual proof involves another lemma called Bézout's identity. This states that if x and y are relatively prime integers (i.e. they share no common divisors other than 1) there exist integers r and s such that

Let a and n be relatively prime, and assume that n | ab. By Bézout, there are r and s making

Multiply both sides by b:

The first term on the left is divisible by n, and the second term is divisible by ab which by hypothesis is divisible by n. Therefore their sum, b, is also divisible by n. This is the generalization of Euclid's lemma mentioned above.

The Disquisitiones Arithmeticae has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.